By [1]} there exists, up to a constant multiple, a function \(\tau (x)\)  which satisfies the following equation near \(\infty _+\) ,
\(\Psi (x;z)=\frac{\tau (x+[z])}{\tau (x)}\exp \left(-\sum _{i=1}^\infty x_i z^{-i}\right).\) 
